
Case 1: \(m \le n\) . We exploit the matrix perturbation bound on singular values provided in [1]}, according to which, given any two matrices \(\mathbf {A},\mathbf {B} \in \mathbb {R}^{m \times n}\) , one has, for all \(i=1,\ldots ,\min \lbrace m,n\rbrace =m\) ,
\(|\sigma _i(\mathbf {A})-\sigma _i(\mathbf {A}+\mathbf {B})| \le \sigma _1(\mathbf {B})\,.\) 
Denoting by \(\Delta \alpha \)  a variation of \(\alpha \) , we apply Eq. (REF ) with \(\mathbf {A}=\mathbf {X}(\alpha )\)  and \(\mathbf {B}=\mathbf {X}(\alpha +\Delta \alpha )-\mathbf {X}(\alpha )=\Delta \alpha \left(\mathbf {X}_1 -\mathbf {X}_2\right)\) . Recalling the relation \(\lambda _i(\mathbf {G}(\alpha ))=\sigma _i^2(\mathbf {X}(\alpha ))\)  valid for all \(i=1,\ldots ,m\) , and the fact that \(\sigma _i(\mathbf {X}(\alpha )) \ge 0\) , we get
\(&&|\lambda _i(\mathbf {G}(\alpha +\Delta \alpha ))-\lambda _i(\mathbf {G}(\alpha ))| \nonumber \\&=&|\sigma _i^2(\mathbf {X}(\alpha +\Delta \alpha ))-\sigma _i^2(\mathbf {X}(\alpha ))| \nonumber \\&=&|\sigma _i(\mathbf {X}(\alpha +\Delta \alpha ))-\sigma _i(\mathbf {X}(\alpha ))|\,(\sigma _i(\mathbf {X}(\alpha +\Delta \alpha ))+\sigma _i(\mathbf {X}(\alpha ))) \nonumber \\&\le & |\Delta \alpha | \,\sigma _1(\mathbf {X}_1-\mathbf {X}_2)\, (\sigma _i(\mathbf {X}(\alpha +\Delta \alpha ))+\sigma _i(\mathbf {X}(\alpha )))\,.\) 
Moreover, \(0 \le \sigma _1(\mathbf {X}_1-\mathbf {X}_2) \le \sigma _1(\mathbf {X}_1)+\sigma _1(\mathbf {X}_2)\)  by Eq. (REF ) with \(\mathbf {A}=\mathbf {X}_1\)  and \(\mathbf {B}=-\mathbf {X}_2\) , whereas \(0 \le \sigma _i(\alpha \mathbf {X}_1+(1-\alpha )\mathbf {X}_2) \le \alpha \sigma _i(\mathbf {X}_1)+(1-\alpha ) \sigma _1(\mathbf {X}_2)\le \sigma _1(\mathbf {X}_1)+\sigma _1(\mathbf {X}_2)\) , where the second last inequality is obtained again by Eq. (REF ) with \(\mathbf {A}=\alpha \mathbf {X}_1\)  and \(\mathbf {B}=(1-\alpha )\mathbf {X}_2\) . Combining all the above, one gets
\(|\lambda _i(\mathbf {G}(\alpha +\Delta \alpha ))-\lambda _i(\mathbf {G}(\alpha ))| \le 2 |\Delta \alpha | \left(\sigma _1(\mathbf {X}_1)+\sigma _1(\mathbf {X}_2)\right)^2\,,\) 
from which one obtains Eq. (REF ) for \(|\Delta \alpha |=|\alpha _k-\alpha | \le \frac{1}{K}\) .

Case 2: \(m > n\) . The proof is the same as above for all but the last (smallest) \(m-n\)  eigenvalues of \(\mathbf {G}(\alpha )\)  and \(\mathbf {G}(\alpha _k)\) . However, the latter eigenvalues are all equal to 0, and the bound (REF ) still holds trivially for them.

